Article ID Journal Published Year Pages File Type
4620338 Journal of Mathematical Analysis and Applications 2009 21 Pages PDF
Abstract

We study the dynamics and regularity of the level sets in solutions of the semilinear parabolic equationut−Δpu+f∈aH(u−μ)in Q=Ω×(0,T],p∈(1,∞), where Ω⊂RnΩ⊂Rn is a ring-shaped domain, ΔpuΔpu is the p-Laplace operator, a and μ   are given positive constants, and H(⋅)H(⋅) is the Heaviside maximal monotone graph: H(s)=1H(s)=1 if s>0s>0, H(0)=[0,1]H(0)=[0,1], H(s)=0H(s)=0 if s<0s<0. The mathematical models of this type arise in climatology, the case p=3p=3 was proposed and justified by P. Stone in 1972. We establish the conditions on the initial data which guarantee that the level sets Γμ(t)={x:u(x,t)=μ} are hypersurfaces, study the regularity of Γμ(t)Γμ(t) and derive the differential equation that governs the dynamics of Γμ(t)Γμ(t). The analysis is based on the introduction of a system of Lagrangian coordinates that transforms the moving surface Γμ(t)Γμ(t) into a stationary one.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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